Ants puzzle

Hi,

You may be familiar with this puzzle.

Consider a regular polygon with n sides, and side length 'a'.There's an ant at each vertex. Every ant moves in one direction(all left or all right) towards the nearest ant in that direction, all at the same,constant speed. They move until they meet. What is the distance traveled by each ant?

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

Re: Ants puzzle

Hi bobbym,

Thanks for the link.I knew of the formula, and but cannot recall the site from where I read. That site too only cited the formula.I tried hard to derive the formula, but couldn't. Then I posted it here!I'm hoping someone may work it out here

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

Re: Ants puzzle

Consider three ants at the end of an equilateral triangle of side length = s moving towards each other at constant speed = v.

As the ants move towards each other, points joining the three ants will be an equilateral triangle of a smaller side.(by symmetry)As time passes the side of this equilateral triangle decreases.

If you choose any of the ant as your frame of reference, the component of velocity of the other ants towards the fixed (frame) ant will be v + v/2. where v is the speed of the ant. [because velocity of A wrt. B is equal to velocity of A wrt. F + velocity of F wrt. B].This means that the fixed ant will see the others coming closer to it at speed = 3/2v.

And in the ant frame the other ants travel towards the fixed one in a straight line. Therefore distance traveled(in ant frame) = s.

Time for which the ants move is = T = dist/speed = s/(3v/2).

In the ground frame they move with speed v for a time = T. Therefore distance traveled by the ants = vT = (2/3)s

Re: Ants puzzle

That same logic can be applied to an n sided polygon. I have not seen any shorter method for this puzzle.The above solution is similar to that given in the book 'Concepts of Physics' by H.C. Verma. Have you seen that book?